Learning the dynamics of shape is at the heart of many computer vision problems: object tracking, change detection, longitudinal shape analysis, trajectory classification, etc. In this work we address the problem of statistical inference of diffusion processes of shapes. We formulate a general Ito diffusion on the manifold of deformable landmarks and propose several drift models for the evolution of shapes. We derive explicit formulas for the maximum likelihood estimators of the unknown parameters in these models, and demonstrate their convergence properties on simulated sequences when true parameters are known. We further discuss how these models can be extended to a more general non-parametric approach to shape estimation.